CONSTRUCTION A CORING FROM TENSOR PRODUCT OF BIALGEBRA

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1 Vol. 7, No. 1, Juni 2012 ONSTUTION OING FOM TENSO PODUT OF ILGE Nikken Prima Puspita, Siti Khabibah Department of Mathematics, Faculty of Mathematics and Natural Sciences, Diponegoro University, Jl. Prof. Soedharto, Semarang, Indonesia nikkenprima@yahoo.com bstract In this Paper introduced a coring from tensor product of bialgebra. n algebra with compatible coalgebra structure are known as bialgebra. For any bialgebra we can obtained tensor product between and itself. Defined a right and left -action on the tensor product of bialgebra such that we have tensor product of and itself is a bimodule over. In this note we expect that the tensor product and itself becomes a -coring with comultiplication and counit. Keywords : action, algebra, coalgebra, coring. bstrak Pada tulisan ini dikenalkan koring dari hasil kali tensor sebuah bialjabar. ljabar yang sekaligus juga koajabar disebut bialjabar. Untuk setiap bialjabar dapat dibentuk hasil kali tensor dari dengan dirinya sendiri. Didefinisikan -aksi kiri dan kanan pada hasil kali tensor tersebut sedemikian hingga hasil kali tensor dengan dirinya sendiri merupakan modul atas. Lebih lanjut hasil kali tensor akan membentuk struktur -koring yang dilengkapi dengan komultiplikasi dan kounit. Kata Kunci : aksi, aljabar, koaljabar, koring ` 1. INTODUTION, -bimodule (left unital). We expect Let is commutative ring. In this paper,, will denoted - module which is assosiative - algebra with multiplication and unit 1 that by define a comultiplication and counit for, -coring. becomes a efore continue to the main as well as coassosiative coalgebra with comultiplication and counit. s a -module, we have tensor product and itself i.e. (Hungerford, 1978). For a left and right action on over assosiative (with unit 1) algebra, can be considered as result, on section 2 we will be given basic concepts about tensor product (Hungerford, 1978). On section 3 we will be given review about algebra, coalgebra and bialgebra (rzeziński, T., Wisbauer,., 2003]). oring is generalized from coalgebra over any ring (associative algebra, not need com- 65

2 onstruction oring...(nikken Prima Puspita, Siti Khabibah) mutative). Section 4 is the main result in this paper. Section 4 contained an explanation about coring from a tensor product of bialgebra. The author assume that readers have been understanding about module theory. This paper has been presented on Seminar Nasional ljabar 2011 Padjajaran University pril 30, 2011 in andung. 2. TENSO PODUT In this section we will review about tensor product. It is important to know about bilinear and balanced map before we study tensor product from The next definition is about tensor product for M and N. Definition 2.2 Let right -module M, left -module N and belian Group G,. belian group M N with bilinear and balanced function is called tensor product from M and N provided for every bilinear and balanced map : M N G there is a unique map : M N G such that diagram above is cummutative diagram, i. e. modules. Definition 2.1 M N G Let right -modue M, left -moduel N and belian Group G,. map M N G is called, : -bilinear M N and balanced function provided for any m, m M, n, n N and r (i). m m, n m, n m, n (ii). m, n n m, n m, n (iii).,, properties) m r n m rn (alance Figure 1. Tensor Product Diagram Notation for element of M N is m i n m i I i i M dan n N i. Theorem 2.3 Let right -module M and left - module N. Tensor product from and N is unique. M 66

3 Vol. 7, No. 1, Juni 2012 Lemma 2.4 If M N is tensor product from right -module M and left -module N, then for any m1, m2 M, n1, n2 N and r (i). m m n m n m n m n n m n m n (ii) (iii). mr n mrn (iv). m0 0n 0 Moreover based on tensor product definition, the isomorphism of tensor product will be given on the theorem above. Theorem 2.8 For every right -module and left -module, then. Theorem 2.9 dan For every right -module,, S - bimodule and left S -module, then. S S Theorem 2.10 For every right -module collection and left -module,. Definiton 2.11 For any right -module and left - module, the twist map is defined as a map tw :, a b b a. The twist map on Definition 2.11 denoted by tw. 3. LGE, OLGE ND ILGE This section must well known before we study topics in this paper. There are many author write about algebra and coalgebra so it is simple to find definition and some properties about algebra and coalgebra. Definition 3.1 Let is a commutative ring. n - algebra is a -module (with 1) such that it must have multiplication :, a b ab. Definition 3.2 (i). n -algebra with multiplication is called associative algebra I I. provided 67

4 onstruction oring...(nikken Prima Puspita, Siti Khabibah) (ii). n -algebra with multiplication is called algebra with unit if there is homomorphism : such that I I. linear map : such that the following diagram is commute. Definiton 3.3 Let and be -algebra. map f : is called an -algebra I I homomorphism if (i). f is a ring homomorphism and f 1 1. (ii). f is a module homomorphism. In this paper an associative - algebra with unit is denoted by,,. easearch about algebra and coalgebra started in 1960 by Sweedler. olagebra is dual from algebra, it can obtained by reverse the algebra multiplication arrow. The following definition and properties are explain the coalgebra structure. Definition 3.4. Let commutative ring and -module. -module is called coalgebra provided it has a -linear map :, c c1 c2 and - Figure 2. ounit Diagram -linear map on Definition 3.4 is called comultiplication and : is called counit map. For any c, c c c c c Definition 3.5 omultiplication : is called coassosiative provided the following diagram is commute. Figure 3. oassosiative Diagram From Definition 3.5 -coalgebra becomes a coassositaive coalgebra provided for any c, I I c c c c c c Throughout in this paper -coalgebra 68

5 Vol. 7, No. 1, Juni 2012 over comultilication and counit denoted by triple,,. Definition 3.6 oring is -coaljabar over any ring (not need commutative). Definition 3.7 Let,,, ', ', ' and - module homomorphism f : '. Map f : ' is called coalgebra morphism provided f f f '. ased on Definition 3.4 and 3.5, any -algebra can be considered as a coassosiative coalgebra, because we can defined comultiplication :, a a a. If there is a set which is an assosiative algebra as well as coassosiative coalgebra, then it can be a bialgebra. Definition 3.8 n -module that is an algebra,, and a coalgebra,, called a bialgebra if and are algebra morphism or, equivantely, and are coalgebra morphism. Definition 3.8, For to be an algebra morphism it means that is I and I tw I. Similarly, is an algebra morphism if and only if I and. I 4. OING FOM IL- GE Throughout this section,, will denoted an associative -algebra with multiplication and unit 1 as well as a coassociative coalgebra with comultiplication and counit, such that a b ab a b,. We always have canonical multiplication with unit 1 1 on and we will defined comultiplication with counit on over associative -algebra. For this aim, bimodule. Lemma 4.1 must be a, - Given -bialgebra,,. Tensor product is a, -bimodule with right and (unital) left -actions with unit 1, i.e 69

6 onstruction oring...(nikken Prima Puspita, Siti Khabibah) :, a, b c ab c :, a b c a bc ac1 bc2 Proof :,. From tensor product properties and are well defined and its easy to proof the modules axioms in. For any a b and 1, 1. a b 1 a b a b. Theorem 4.2 For -bialgebra,, define a maps :.1, ab ab 1b a1 b 1 b, I : ab ab ab a b a b Then maps is coassosiative weak comultiplication on the left and the is -module morphism with a b I a b.1, for all a, b. Proof: For a, b, c we have c. a b ca b ca b 1b 1 2 ca1 b 1 b c a1 b 1 b c a1 b 1 b c. a b 1 b. c c1 bc c1 bc2 1 bc2 c11 b1c 2112 b2c 22 (from coassosiativity ) 1 2 c b c b c b. c 1 b1 1 b2. c (1) 1 b1 c1 b2c2 1 b1. c1 b2c2 1 b1 c1 b2c2 c1 1 b1c1 2 b2c2 I a b I ab b ab1 1b2 ab1 111 b211 2 b22 a111 b11 12 b211 2 b22 a1 b b b2 a1 b b12.1b2 a1 b b12 1b2 ab1 1b2 I ab b I ab ased on ommutative diagram in Figure 3 is a coassosiative weak comultiplication on. learly is left -linear and 70

7 Vol. 7, No. 1, Juni 2012 I 1 1 a b I a b b2 a b1 1b2 a b1 1 b2.1 a b1 11 b212 a1 1 1 b b212 a1 b 1 b a 1 b 1 12 a b.1 (from counital ) nalog for I a b a b.1. In general properties from and are not sufficient to make be a coring. Map need neither be right -linear nor I a b a b.1. To ensure these properties we have to pose additional conditions on and. Theorem 4.3 Triple,, induces a -coring structure on if only if is a bialgebra with ab a b, all a, b and Proof: for ssume is a bialgebra with ab a b, a, b and For a, b, c, then is unital, -bimodule a b.1 a b 1 a b 1 1 a b, and a b. c ac1 bc2 properties) ac bc a b c c ( a b c (counital ) a1 b 1. c a b. c showing that is right -linear and by Theorem 4.2 so ssume is -coring. is -coring, then is a unital right -module , is right -linear so we have 1. 1 and a b a b 11 a1 b a b a. b b1 ab2 b1 ab2 by applying properties of we get b1 ab2 a b1 b2 ab ab a b. ab b a b b EFEENES rzeziński, T., Majid, Sh. 1998, oalgebra undles, omm. Math. Phys, 191 : rzeziński, T The cohomology structure of an algebra entwined 71

8 onstruction oring...(nikken Prima Puspita, Siti Khabibah) with coalgebra, Jurnal of lgebra 235 : rzeziński, T., Wisbauer, oring and comodules, Germany. rzeziński, T. The Structures of orings, lg. ep Theory, to appear. Hungerford, T.W lgebra, Graduate text in Mathematics, Springer-Verlag, erlin. Puspita, N. P Koring Lemah, Thesis, Gadjah Mada University, Yogyakarta. Wisbauer, Foundation of Module and ing Theory, Gordon and reach Science Publishers, Germany. Wisbauer,., Weak oring, Jurnal of lgebra 245 :